Solve PDE using method of characteristics with non-local boundary conditions.
Given the population model by the following linear first order PDE in u(a,t) with constants b and :
We can split the integral in two with our non-local boundary data:
Choosing the characteristic coordinates and re-arranging the expression to form the normal to the solution surface we have the following equation with initial conditions:
Solving each of these ODE's in gives the following:
Substituting back the original coordinates we can re-write this expression with a coordinate change:
Now this is where I get stuck, how do I use the boundary data to come up with a well-posed solution?